Open Conjectures
on bricks in representation theory of algebras
Throughout, k is an algebraically closed field and A = kQ/I is a basic, connected, finite-dimensional associative k-algebra. An A-module M is a brick if EndA(M) is a division algebra (equivalently, EndA(M) ≅ k); A is brick-infinite if it has infinitely many isomorphism classes of bricks — equivalently, if it is τ-tilting infinite. The conjectures below are listed in the chronological order of their first appearance in the literature (by arXiv date, or thesis date where no arXiv preprint exists), and are — to the best of current knowledge — still open in general.
1. g-vector(s) Conjecture
Setting: the τ-tilting fan of A is the fan in Rn formed by the g-vector cones of the basic support τ-tilting A-modules.
A is brick-infinite if and only if there exists a rational ray in Rn lying outside every cone of the τ-tilting fan of A.
Now treated as a conjecture, the statement was first raised as an open problem (Question 3.49) by Laurent Demonet in his 2017 Habilitation thesis, in the context of the τ-tilting fan — the fan formed in Rn by the g-vector cones of the basic support τ-tilting modules of an algebra. The question ties brick-infiniteness to a geometric "gap" in this fan.
Methodology and partial results: Calvin Pfeifer (arXiv:2308.09576, August 2023) gave a τ-reduced reformulation of the conjecture in terms of Geiss–Leclerc–Schröer's generically τ-reduced components, and proved it equivalent to the Stability Conjecture for the class of E-tame algebras introduced by Asai–Iyama. Shortly after, Mousavand–Paquette (arXiv:2311.14863, November 2023) introduced the τ-convergence property as a systematic method for constructing a rational g-vector outside the τ-tilting fan: for a minimal brick-infinite algebra with this property, such a g-vector always exists, so the algebra cannot be E-finite. Using this method, they verify the conjecture for every tame algebra all of whose bricks are, with finitely many exceptions, τ-rigid.
Current state: open in general.
Connections: part of the unifying brick-Brauer-Thrall (bBT) framework; equivalent to the Stability Conjecture for E-tame algebras, and to all bBT conjectures for tame algebras in general.
Related topics
2. Second brick-Brauer-Thrall (2nd bBT) Conjecture
Setting: a finite-dimensional algebra A which is brick-infinite.
If A is brick-infinite, there exists at least one positive integer d such that A admits infinitely many pairwise non-isomorphic bricks of length (k-dimension) d.
First posed by Kaveh Mousavand as Conjecture 6.6 of the arXiv preprint "τ-tilting finiteness of non-distributive algebras and their module varieties" (October 2019), this can be seen as the brick-theoretic analogue of the classical Second Brauer–Thrall Conjecture -- proved in the 1980s. Later, the same statement was independently posed by Schroll–Treffinger–Valdivieso under the name "Second τ-Brauer-Thrall Conjecture."
Current state: verified for several families — non-distributive minimal representation-infinite algebras, biserial algebras, and any algebra with a generalized standard component (hence tame algebras, under a standard reduction) — but open in general.
Connections: the central conjecture of the bBT family; it implies the Rigid-bricks and Semibrick conjectures, and reduces to the case of minimal brick-infinite algebras with almost all bricks faithful.
Related topics
3. Rigid-bricks Conjecture
Setting: a brick X is rigid if Ext1A(X, X) = 0.
If every brick of A is a rigid module, then A is brick-finite.
Posed alongside the 2nd bBT Conjecture in Mousavand's 2019–2020 work (Conjecture 6.0.1 of the Ph.D. thesis). A geometric argument shows that an affirmative resolution of the 2nd bBT Conjecture would also settle this one.
Current state: open in general; verified in the same special cases as the 2nd bBT Conjecture.
Connections: a consequence of the 2nd bBT Conjecture; both sit inside the wider bBT family unified through Hom-orthogonal modules.
Related topics
4. Semibrick Conjecture
Setting: a semibrick is a set of bricks that is pairwise Hom-orthogonal, i.e. Hom(X, Y) = Hom(Y, X) = 0 for any two distinct members X, Y.
A is brick-infinite if and only if A admits an infinite semibrick.
First conjectured by Haruhisa Enomoto as Conjecture 5.12 of the arXiv preprint "Monobrick, a uniform approach to torsion-free classes and wide subcategories" (May 2020), building on the classical bijection between semibricks and wide subcategories in a length abelian category. The conjecture is posed in that setting.
Current state: open in general; known to hold for biserial algebras and for any algebra with a generalized standard component, hence for tame algebras under the standard reduction.
Connections: shown to follow from the 2nd bBT Conjecture (Mousavand–Paquette, 2024); equivalent to the other bBT conjectures for tame algebras.
Related topics
5. Generic-brick Conjecture
Setting: an indecomposable A-module G is generic if it has infinite k-dimension but finite length over EndA(G) (finite endolength); G is a generic brick if, in addition, EndA(G) is a division ring.
A is brick-infinite if and only if A admits a generic brick.
The conjecture rests on a foundational dichotomy between finite- and infinite-dimensional bricks: Francesco Sentieri (arXiv:2011.09253, November 2020) proved that A is τ-tilting finite if and only if every brick over A — allowing infinite-dimensional ones — is finitely generated. Equivalently, A is brick-infinite if and only if it admits an infinite-dimensional brick, which is what makes the search for a generic brick (a particularly well-behaved kind of infinite-dimensional brick) a natural approach to detecting brick-infiniteness.
Originally formulated by Mousavand–Paquette as Conjecture 1.1 in "Biserial algebras and generic bricks" (arXiv:2209.05696, September 2022), building on Sentieri's theorem to extend Crawley-Boevey's classical theory of generic modules for tame algebras to the brick setting. Bautista–Pérez–Salmerón (arXiv:2408.16127, August 2024, still a preprint) later proved that, for a tame algebra, a generic module is a generic brick if and only if it determines a one-parameter family of bricks of the same dimension — equivalently, a tame algebra admits a generic brick if and only if it is brick-continuous. Most recently, Kevin Schlegel (arXiv:2606.23297, June 2026) gave a further characterization: over a tame algebra, a generic module is τ--rigid if and only if it is a brick, linking the conjecture to infinite τ-tilting theory.
Current state: open in general; verified for biserial algebras. For tame algebras, the above characterizations pin down exactly when a generic module is a generic brick, reducing the conjecture to a concrete existence question about generic modules.
Connections: equivalent to the other bBT conjectures for tame algebras; for a tame algebra, all bBT conjectures hold if and only if it admits a homogeneous brick.
Related topics
6. Stability Conjecture
Setting: for θ ∈ K0(proj A), a module is θ-stable in the sense of King's stability condition with respect to θ.
A is brick-infinite if and only if, for some θ ∈ K0(proj A), there are infinitely many pairwise non-isomorphic θ-stable modules of the same dimension vector.
Traces back to a 2012–2013 conjecture of Chindris–Kinser–Weyman on the "dense orbit property" and Schur-representation-finiteness, combined with a related 2019 conjecture of Mousavand. The combined statement, phrased in terms of King's stability condition, was formulated by Calvin Pfeifer (arXiv:2308.09576, August 2023), who also connected it to a geometric reformulation of the g-vectors Conjecture (see item 1 above) and proved this conjecture implies that reformulation, with the converse holding for E-tame algebras.
Progress on E-tame algebras: shortly after, and independently of Pfeifer, Mousavand–Paquette (arXiv:2311.14863, November 2023) prove that an E-tame algebra which is not E-finite always admits an infinite family of bricks of the same dimension — the statement of this conjecture restricted to the E-tame case. They also show that an E-tame algebra all of whose bricks are, with finitely many exceptions, τ-rigid must be brick-finite, extending their tame-algebra result (see item 1 above) to the broader E-tame setting.
Current state: open in general. For E-tame algebras, Pfeifer's equivalence reduces the conjecture to the g-vectors Conjecture (item 1), and Mousavand–Paquette's result supplies the conjecture's conclusion once an algebra is known not to be E-finite — but whether every brick-infinite E-tame algebra fails to be E-finite is exactly the content of the (still open) g-vectors Conjecture for that class.
Connections: part of the unifying bBT framework; equivalent to the other bBT conjectures for tame algebras.
Related topics
7. Hom-orthogonal Conjecture
Setting: a family {Xi} of modules is pairwise Hom-orthogonal if Hom(Xi, Xj) = Hom(Xj, Xi) = 0 for all i ≠ j; A has rank n (its number of simple modules).
For an algebra A of rank n, the following are equivalent:
(1) mod A contains an infinite family of pairwise Hom-orthogonal modules of the same dimension;
(2) mod A contains n + 1 pairwise Hom-orthogonal bricks;
(3) A is brick-infinite.
(The implications (1) ⇒ (2) ⇒ (3) are known; the converses are open in general.)
Introduced by Mousavand–Paquette in "Hom-orthogonal modules and brick-Brauer-Thrall conjectures" (July 2024), as an elementary, purely module-theoretic reformulation designed to unify the entire bBT family into a single statement.
Current state: the implications (1) ⇒ (2) ⇒ (3) are proved in general; the reverse implications are open in general but known to hold for tame algebras, where all three conditions become equivalent.
Connections: the main technical tool behind the proof that the g-vectors, 2nd bBT, Semibrick, Generic-brick, and Stability conjectures all become equivalent for tame algebras.
Related topics